SETS QUESTIONS (With detailed solutions)

Sets Questions (A-F)

Question A

(a) Simplify each of the following set operations using the laws of algebra of sets:

(i) A - (A² - B)

(ii) [A ∩ (A² ∪ B)] ∪ [B ∩ (A² ∩ B²)]

(b) Given:

A = {X : X ∈ Z⁺, X ≤ 10}
B = {X : X ∈ Z⁺, X is even and X ≤ 10}
μ = {X : X ∈ Z⁺, X ≤ 20}

Find:
i. A - B
ii. A² ∩ B²
iii. A² ∪ B

(c) At a senior academic there are 40 people:

- 20 are men
- 18 take mathematics
- 6 of the women are not taking mathematics
- 4 of the men are not taking chemistry
- 13 are taking chemistry out of 18 who are taking mathematics

Evaluate:
i. The number of women taking both subjects
ii. The number of men who take mathematics

Question B

(a) Set A is defined as:

A = {x² : x ∈ z⁺ and 0 ≤ x < 6}

Describe set A by roster form

(b) Given universal set U and subsets A, B:

U = {a, b, c, d, e, f, g, h, i, j}
A ∩ U = {d, e, h, i}
A ∩ B = {d, h}
(A ∪ B)* = {f, g, j}

i. Represent the above information in a Venn diagram
ii. Find the number of elements of sets B and A' ∩ B

Question C

In a class of 58 students:

- 30 are boys and the rest are girls
- 29 study mathematics
- 27 study physics
Among the girls:
- 3 study both mathematics and physics
- 9 study mathematics only
- 4 study neither mathematics nor physics
Among the boys:
- 5 do not study either of these two subjects

a. Summarize the information using a Venn diagram
b. How many students study both subjects?
c. How many boys study physics only?

Question D

60 students study Physics, Chemistry and Mathematics:

- 12 students score A in both Physics and Chemistry
- Half of the students score A in Mathematics
- One third score A in Chemistry
- 5% score A in Physics only
- 17 students don't have A in any subject

How many score A in:
i. All three subjects
ii. Chemistry only

Question E

A poultry farm's six-month report revealed:

- 127 regular customers
- 66 bought broilers
- 81 bought layers
- 76 bought cocks
- 46 bought layers and cocks
- 34 bought broilers and cocks
- 11 bought broilers only
- 15 bought layers only
- 6 bought cocks only
- 6 customers did not show up

(a) How many customers bought all three products?
(b) How many customers bought exactly two of the farm's products?

Question F

A group of students consists of 17 girls and 15 boys:

- 22 play handball
- 16 play basketball
Among the boys:
- 12 play handball
- 11 play basketball
- 10 play both games
Among the girls:
- 3 do not play either of the games

(a) Summarize the given information using a Venn diagram
(b) How many students play either handball or basketball?
(c) How many girls play both games?

Sets Questions Solutions

Question A

(a) Simplify each of the following sets operation:

(i) A - (A² - B)

1. First, understand that A² is likely a typo and should be A' (complement of A)

2. Rewrite the expression: A - (A' - B)

3. Using set difference: X - Y = X ∩ Y'

4. So A' - B = A' ∩ B'

5. Now A - (A' ∩ B') = A ∩ (A' ∩ B')'

6. Apply De Morgan's Law: (A' ∩ B')' = A ∪ B

7. Final expression: A ∩ (A ∪ B) = A (by absorption law)

Answer: A

(ii) [A ∩ (A² ∪ B)] ∪ [B ∩ (A² ∩ B²)]

1. Assuming A² = A' and B² = B'

2. First part: A ∩ (A' ∪ B) = (A ∩ A') ∪ (A ∩ B) = ∅ ∪ (A ∩ B) = A ∩ B

3. Second part: B ∩ (A' ∩ B') = B ∩ A' ∩ B' = A' ∩ (B ∩ B') = A' ∩ ∅ = ∅

4. Combine: (A ∩ B) ∪ ∅ = A ∩ B

Answer: A ∩ B

(b) Given sets:

• A = {X : X ∈ Z⁺, X ≤ 10} = {1,2,3,4,5,6,7,8,9,10}
• B = {X : X ∈ Z⁺, X is even and X ≤ 10} = {2,4,6,8,10}
• μ = {X : X ∈ Z⁺, X ≤ 20} (universal set)

i. A - B

1. A - B means elements in A that are not in B

2. A has all integers 1-10, B has even numbers 2-10

3. Therefore, A - B = {1,3,5,7,9}

Answer: {1,3,5,7,9}

ii. A² ∩ B²

1. A² = A' = μ - A = {11,12,...,20}

2. B² = B' = μ - B = {1,3,5,7,9,11,12,...,20}

3. A' ∩ B' = {11,12,...,20} (only elements in both complements)

Answer: {11,12,13,14,15,16,17,18,19,20}

iii. A² ∪ B

1. A² = A' = {11,12,...,20}

2. B = {2,4,6,8,10}

3. A' ∪ B = {2,4,6,8,10,11,12,...,20}

Answer: {2,4,6,8,10,11,12,...,20}

(c) Venn Diagram Problem

Given information:

• Total people: 40
• Men: 20 → Women: 20
• Taking Mathematics: 18
• Women not taking Mathematics: 6
• Men not taking Chemistry: 4
• Taking Chemistry: 13
• Of 18 taking Mathematics, 13 are taking Chemistry

i. Number of women taking both subjects

1. Women not taking Math: 6 → Women taking Math: 20 - 6 = 14

2. Total taking Math: 18 → Men taking Math: 18 - 14 = 4

3. Of Math takers, 13 take Chemistry → 5 Math takers don't take Chemistry

4. Women taking both: Let x = women taking both

5. Then women taking only Math: 14 - x

6. Total Chemistry takers: 13 = (men taking both) + x

7. Men not taking Chemistry: 4 → Men taking Chemistry: 20 - 4 = 16

8. But only 4 men take Math, so max men taking both is 4

9. Therefore, x = 13 - 4 = 9

Answer: 9 women take both subjects

ii. Number of men who take mathematics

1. From above, total Math takers: 18

2. Women taking Math: 14

3. Therefore, men taking Math: 18 - 14 = 4

Answer: 4 men take mathematics